Quasi-Steady State Approximation for Partial Differential Equations

• Applicant: Professor Christian Kühn, Ph.D.
• Subject Area: Mathematics
• Term: from 2021 to 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 456754695

• Final Report Year: 2025

Final Report Abstract

Many systems in science and engineering involve processes occurring at vastly different time scales. The Quasi-Steady State Approximation (QSSA) is a method to simplify such systems by focusing on the slower dynamics and treating the faster ones as nearly equilibrated. While this method is well-established for ordinary differential equations (ODEs), its rigorous application to partial differential equations (PDEs) — which also include spatial components — remains a frontier of research. This project aimed to advance the theoretical foundation of QSSA for PDEs by combining analytical methods, such as entropy-based approaches, with geometric techniques like slow manifold reductions. A major focus was the derivation of simplified models that retain essential dynamical features of the full systems but are more tractable for analysis and computation. Key case studies included reaction-diffusion systems from chemistry, ecological predator-prey models, and stochastic systems represented by the Fokker-Planck equation. The project demonstrated how complex multiscale models can be rigorously reduced to simpler equations without losing critical information, providing insight into spatial dynamics and stability of such systems. These findings are significant because they allow researchers and practitioners to analyze and simulate complex processes more efficiently and accurately. The project also had strong outreach and training components, involving early-career researchers and students, thus contributing to capacity building in mathematical sciences.

Publications

• Global well-posedness for volume–surface reaction–diffusion systems. Communications in Contemporary Mathematics, 25(04).
  Morgan, Jeff & Tang, Bao Quoc
  
• Analysis of mass controlled reaction–diffusion systems with nonlinearities having critical growth rates. Journal of Evolution Equations, 23(3).
  Sun, Chunyou; Tang, Bao Quoc & Yang, Juan
  
• Global Renormalised Solutions and Equilibration of Reaction–Diffusion Systems with Nonlinear Diffusion. Journal of Nonlinear Science, 33(4).
  Fellner, Klemens; Fischer, Julian; Kniely, Michael & Tang, Bao Quoc
  
• Fast-reaction limits for predator–prey reaction–diffusion systems: improved convergence. Topics in Multiple Time Scale Dynamics, 173–187.
  Soresina, Cinzia; Tang, Bao & Tran, Bao-Ngoc
  
• Infinite dimensional slow manifolds for a linear fast-reaction system. Contemporary Mathematics, 87-104. American Mathematical Society.
  Kuehn, Christian; Lehner, Pascal & Sulzbach, Jan-Eric
  
• Macroscopic limit for stochastic chemical reactions involving diffusion and spatial heterogeneity. Stochastic Processes and their Applications, 176, 104433.
  Egan, Malcolm & Tang, Bao Quoc
  
• Nonconcentration phenomenon for one‐dimensional reaction–diffusion systems with mass dissipation. Mathematische Nachrichten, 297(11), 4288-4306.
  Yang, Juan; Kostianko, Anna; Sun, Chunyou; Tang, Bao Quoc & Zelik, Sergey
  
• Rigorous Derivation of Michaelis–Menten Kinetics in the Presence of Slow Diffusion. SIAM Journal on Mathematical Analysis, 56(5), 5995-6024.
  Tang, Bao Quoc & Tran, Bao-Ngoc
  
• Approximate slow manifolds in the Fokker-Planck equation. Quarterly of Applied Mathematics, 84(2), 227-261.
  Kuehn, Christian & Sulzbach, Jan-Eric
  
• Fast reactions and slow manifolds. Nonlinear Differential Equations and Applications NoDEA, 32(4).
  Kuehn, Christian & Sulzbach, Jan-Eric
  
• Nonisothermal Diffuse Interface Model for Phase Transition and Interface Evolution. SIAM Journal on Applied Mathematics, 85(6), 2611-2635.
  Liu, Chun; Sulzbach, Jan-Eric & Wang, Yiwei
  
• On a continuum model for random genetic drift: a dynamic boundary condition approach. Communications in Analysis and Mechanics, 17(3), 829-848.
  Liu, Chun; Sulzbach, Jan-Eric & Wang, Yiwei
  
• Pathwise Mild Solutions for Superlinear Stochastic Evolution Equations and Their Attractors. Elsevier BV.
  Neamtu, Alexandra; Seitz, Tim; Sonner, Stefanie & TANG, Bao Quoc
  
• Slow Manifolds for PDE with Fast Reactions and Small Cross Diffusion
  L. Desvillettes, C. Kuehn, J.E. Sulzbach, B.Q. Tang & B.N. Tran
  
• Volume-surface systems with sub-quadratic intermediate sum on the surface: Global existence and boundedness. Communications on Pure and Applied Analysis, 24(12), 2261-2289.
  Yang, Juan & Tang, Bao Quoc
  
• On the Equilibriation of Chemical Reaction-Diffusion Systems with Degenerate Reactions. SIAM Journal on Mathematical Analysis, 58(1), 276-307.
  Desvillettes, Laurent; Phung, Kim Dang & Tang, Bao Quoc
  
• Rigorous fast signal diffusion limit and convergence rates with the initial layer effect in a competitive chemotaxis system. Elsevier BV.
  Reisch, Cordula; Tran, Bao-Ngoc & Yang, Juan
  
• Singular limit and convergence rate via projection method in a model for plant-growth dynamics with autotoxicity. Journal of Differential Equations, 452, 113797.
  Morgan, Jeff; Soresina, Cinzia; Tang, Bao Quoc & Tran, Bao-Ngoc